Question

polynomial division box method (guided)
score: 0/2 penalty: 1 off
question
fill in the missing values below one at a time to find the quotient when $9x^3 - 6x^2 - 29x - 14$ is divided by $3x + 2$.
box with missing values, first row first cell empty, second row: 3x, 9x³, empty, empty; third row: +2, empty, empty, empty
answer attempt 1 out of 2
you must answer all questions above in order to submit.

Explanation:

Step1: Find the first term of the quotient

To find the first term of the quotient, we divide the leading term of the dividend \(9x^{3}\) by the leading term of the divisor \(3x\). So, \(\frac{9x^{3}}{3x}=3x^{2}\). This is the value that goes in the top - left box of the box method (the first term of the quotient).

Step2: Multiply the divisor by the first term of the quotient

Now, we multiply the entire divisor \(3x + 2\) by \(3x^{2}\).

• For the \(3x\) part: \(3x\times3x^{2}=9x^{3}\) (which is already given in the box).
• For the \(+2\) part: \(2\times3x^{2}=6x^{2}\). So the box under \(+2\) and in the same column as \(9x^{3}\) should be \(6x^{2}\).

Step3: Subtract and find the next term

Next, we look at the next term of the dividend, which is \(-6x^{2}\). We subtract the term we just found (\(6x^{2}\)) from \(-6x^{2}\): \(-6x^{2}-6x^{2}=-12x^{2}\). Now we divide this new leading term \(-12x^{2}\) by the leading term of the divisor \(3x\): \(\frac{-12x^{2}}{3x}=-4x\). This is the second term of the quotient.

Step4: Multiply the divisor by the second term of the quotient

Multiply \(3x + 2\) by \(-4x\).

• For the \(3x\) part: \(3x\times(-4x)=-12x^{2}\).
• For the \(+2\) part: \(2\times(-4x)=-8x\).

Step5: Subtract and find the next term

Now, look at the next term of the dividend, which is \(-29x\). Subtract the term we just found (\(-8x\)) from \(-29x\): \(-29x-(-8x)=-29x + 8x=-21x\). Divide this new leading term \(-21x\) by the leading term of the divisor \(3x\): \(\frac{-21x}{3x}=-7\). This is the third term of the quotient.

Step6: Multiply the divisor by the third term of the quotient

Multiply \(3x + 2\) by \(-7\).

• For the \(3x\) part: \(3x\times(-7)=-21x\).
• For the \(+2\) part: \(2\times(-7)=-14\) (which should match the last term of the dividend).

First, the top - left box (first term of quotient) is \(3x^{2}\). The box under \(+2\) and in the \(9x^{3}\) column is \(6x^{2}\). The second term of the quotient is \(-4x\), the box for \(3x\) and the \(-12x^{2}\) column is \(-12x^{2}\), the box for \(+2\) and the \(-12x^{2}\) column is \(-8x\), the third term of the quotient is \(-7\), the box for \(3x\) and the \(-21x\) column is \(-21x\), and the box for \(+2\) and the \(-21x\) column is \(-14\). But since the first missing value (top - left box) is what is likely being asked first, we focus on that.

Answer:

The first missing value (top - left box, first term of the quotient) is \(3x^{2}\)